A Study Of Analytical Solution Methods Of Several Kinds Of Nonlinear Integrable Partial Differential Equations

Up to now,researchers usually explore important nonlinear phenomena such as wave physical structures and dynamical behaviors through analytic solutions of nonlinear partial differential equations.This makes the study of the analytical solution method of nonlinear partial differential equations have certain theoretical and practical significance.This dissertation focus on the research of several kinds of methods for analytical solutions to nonlinear partial differential equations and their applications,which is divided into the following three parts:1.The application of Lie symmetry method to nonlinear boundary value problems in fluid mechanics is devoted.The Wu differential characteristic set algorithm is used to obtain a group of multi-parameter symmetries for the boundary value problem,and the Lie symmetry method is adopted to simplify the boundary value problem into some initial value problems of reduced diffusion equations.Approximate analytical and numerical solutions are then obtained by utilizing the homotopy perturbation method and the Runge-Kutta method respectively,and compared to exhibit the convergence of approximate analytical solutions.2.Several classes of(3+1)-dimensional nonlinear evolution equations are generalized.And then,some novel analytical solutions,including lump-type solutions,breather lump wave solutions as well as multiple soliton solutions are successfully constructed,based on the Hirota bilinear method.Particularly,a type of new interactive relationship in terms of a new combination of quadratic function,trigonometric function and exponential function is given,namely,a kind of periodic lump-kink(stripe)wave solution,which is a sort of mixed type solutions of periodic lump wave and multi-kink(stripe)waves.Finally,dynamics characteristics and interaction phenomena are exhibited for the obtained solution waves through particular plots with proper choices of different values for the parameters.3.The trilinear method and its application are given.The multivariate trilinear operators in(3+1)-dimensional space are applied to two(3+1)-dimensional nonlinear evolution equations.The resulting trilinear forms are used to study their wave dynamics.Also,a type of new interaction solution between a breather lump wave and multi-kink waves is generated,namely,a kind of breather luxmp-kink wave solution.In the same way,the interaction phenomena are depicted by plotting 3-D and density graphs from the perspective of wave characteristics.